Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 17 - Second-Order Differential Equations - 17.2 Nonhomogeneous Linear Equations - 17.2 Exercises - Page 1207: 17

Answer

$y_p(x)=xe^{-x}[(A x^2+Bx+C)\cos 3x+(D x^2+Ex+F)\sin 3x]$

Work Step by Step

Consider $G(x)=e^{\alpha x} A(x) \sin mx $ or $G(x)=e^{\beta x} A(x) \cos mx $ The trial solution for the method of undetermined coefficients is defined as: $y_p(x)=e^{\alpha x} B(x) \sin mx +e^{\beta x} C(x) \cos mx$ When the sum of the coefficients of a differential equation is zero. Then, we have $y_p(x)=x e^{\alpha x} B(x)$ On comparing the above equation with the given equation, we get $m=k=1$ Thus, the trial solution for the method of undetermined coefficients is: $y_p(x)=xe^{-x}(A x^2+Bx+C)\cos 3x+xe^{-x}(D x^2+Ex+F)\sin 3x$ Hence, we have $y_p(x)=xe^{-x}[(A x^2+Bx+C)\cos 3x+(D x^2+Ex+F)\sin 3x]$
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